Classical algorithms for quantum mean values

We consider the task of estimating expectation value an $n$-qubit tensor product observable $O_1\otimes O_2\otimes \cdots \otimes O_n$ in output state a shallow quantum circuit. This is cornerstone variational algorithms for optimization, machine learning, and simulation many-body systems. Here we study its computational complexity constant-depth circuits three types single-qubit observables $O_j$ which are (a) close to identity, (b) positive semidefinite, (c) arbitrary. It shown that mean problem admits classical approximation algorithm with runtime scaling as $\mathrm{poly}(n)$ $2^{\tilde{O}(\sqrt{n})}$ cases (a,b) respectively. In case give linear-time geometrically local on two-dimensional grid. The approximated small relative error (a), while (b,c) satisfy less demanding additive bound. based (respectively) Barvinok's polynomial interpolation method, OR function arising from query complexity, Monte Carlo method combined Matrix Product State techniques. also prove technical lemma characterizing zero-free region certain polynomials associated circuit, may be independent interest.